Questions: Determine whether the distribution is a discrete probability distribution. x 0 1 2 3 4 P(x) 0.09 0.25 0.14 0.71 -0.19 Is the distribution a discrete probability distribution? O A. Yes, because the sum of the probabilities is equal to 1. O B. Yes, because the sum of the probabilities is equal to 1 and each probability is between 0 and 1, inclusive. O C. No, because the sum of the probabilities is not equal to 1. O D. No, because not all the probabilities are between 0 and 1, inclusive.

Solution Steps

To determine whether the given distribution is a discrete probability distribution, we need to check two conditions:

1. The sum of all probabilities should be equal to 1.
2. Each probability should be between 0 and 1, inclusive.

We have the probabilities \( P(x) = [0.09, 0.25, 0.14, 0.71, -0.19] \). The sum of these probabilities is calculated as follows:

\[ \text{Sum} = 0.09 + 0.25 + 0.14 + 0.71 - 0.19 = 1.0 \]

Next, we need to verify if each probability \( P(x) \) lies within the range \( [0, 1] \). The probabilities are:

• \( P(0) = 0.09 \) (valid)
• \( P(1) = 0.25 \) (valid)
• \( P(2) = 0.14 \) (valid)
• \( P(3) = 0.71 \) (valid)
• \( P(4) = -0.19 \) (invalid)

Since \( P(4) = -0.19 \) is less than 0, not all probabilities are valid.

Based on the results:

• The sum of the probabilities is \( 1.0 \), which satisfies one condition.
• However, since not all probabilities are between \( 0 \) and \( 1 \), the second condition is violated.

Final Answer